PHYSICAL REVIEW E 98, 052415 (2018)Editors’ Suggestion

Embryonic inversion in Volvox carteri: The flipping and peeling of elastic lips

Pierre A. Haas* and Raymond E. Goldstein†

Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences,University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

(Received 2 August 2018; published 30 November 2018)

The embryos of the green alga Volvox carteri are spherical sheets of cells that turn themselves inside out atthe close of their development through a program of cell shape changes. This process of inversion is a model formorphogenetic cell sheet deformations; it starts with four lips opening up at the anterior pole of the cell sheet,flipping over, and peeling back to invert the embryo. Experimental studies have revealed that inversion is arrestedif some of these cell shape changes are inhibited, but the mechanical basis for these observations has remainedunclear. Here, we analyze the mechanics of this inversion by deriving an averaged elastic theory for these lipsand we interpret the experimental observations in terms of the mechanics and evolution of inversion.

DOI: 10.1103/PhysRevE.98.052415

I. INTRODUCTION

Cell sheet deformations pervade animal development [1]but are constrained by local and global geometry. The localconstraints are the compatibility conditions of differentialgeometry expressed by the Gauß–Mainardi–Codazzi equa-tions [2]. The global constraints by contrast constitute anevolutionary freedom: By evolving different global geome-tries, organisms can alleviate geometric constraints. This ideais embodied in the inversion process by which the differentspecies of Volvox turn themselves inside out at the close oftheir embryonic development.

Volvox [Fig. 1(a)] is a genus of multicellular sphericalgreen algae recognized as model organisms for the evolutionof multicellularity [3–6] and biological fluid dynamics [7]. Anadult Volvox spheroid consists of several thousand biflagel-lated somatic cells and a much smaller number of germ cellsor gonidia [Fig. 1(a)] embedded in an extracellular matrix [3].The germ cells repeatedly divide to form a spherical cell sheet,with cells connected to their neighbors by the remnants ofincomplete cell division, thin membrane tubes called cyto-plasmic bridges [8,9]. Those cell poles whence will emanatethe flagella point into the sphere though, and so the ability toswim is only acquired once the organism turns itself insideout through a hole, the phialopore, at the anterior pole ofthe cell sheet [3,10]. This process of inversion is driven bya program of cell shape changes [11–14]. The key cell shapechange is the formation of wedge-shaped cells with thin stalks[Fig. 1(b)]; at the same time, the cytoplasmic bridges move to

*P.A.Haas@damtp.cam.ac.uk†R.E.Goldstein@damtp.cam.ac.uk

Published by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.

connect the cells at their thin wedge ends [9], thus splayingthe cells and hence bending the cell sheet.

The precise sequence of cell sheet deformations and pro-gram of cell shape changes driving inversion varies fromspecies to species [10] but is broadly classified into twoinversion types [Fig. 1(c)]: In type-A inversion [10,11,13],four lips open up at the anterior pole of the cell sheet, flip over,and peel back to achieve inversion. Type-B inversion [10,14]starts with a circular invagination near the equator of the cellsheet, which initiates inversion of the posterior hemisphere.The phialopore then widens and the anterior hemisphere peelsback over the partly inverted posterior to achieve inversion.

In spite of these differences, the mechanical crux for bothinversion types is the widening of the phialopore to enablethe cell sheet to pass through it. In type-A inversion, this isfacilitated by the presence of the lips, and a simple program ofcell shape changes suffices to invert the cell sheet [11,13]: Thecells of Volvox carteri become spindle-shaped at the beginningof inversion [Fig. 1(b)]. A group of cells near the phialoporethen become flask-shaped, with thin stalks. This bend regionexpands towards the posterior pole, leaving behind column-shaped cells [Fig. 1(b)]. The program of cell shape changesin type-B inversion, by contrast, is rather more complicated,involving different types of cell shape changes in differentparts of the cell sheet [14]; in particular, cells and cytoplasmicbridges near the phialopore elongate in the circumferentialdirection to widen the phialopore.

We have previously analyzed the mechanics of type-Binversion in detail [15–17], because it shares the feature of in-vagination with developmental events in higher organisms [1],but the mechanics of type-A inversion and its lips haveremained unexplored. Additionally, previous studies have re-vealed that type-A inversion in Volvox carteri is arrested (a) ifactomyosin-mediated contraction is inhibited chemically [18]and (b) in a mutant in which the cytoplasmic bridges cannotmove relative to the cells [19]. The precise mechanical basisfor these observations has remained unclear, however.

Here we analyze the mechanics of the opening of thephialopore in type-A inversion by the flip-over of the lips. We

2470-0045/2018/98(5)/052415(11) 052415-1 Published by the American Physical Society

PIERRE A. HAAS AND RAYMOND E. GOLDSTEIN PHYSICAL REVIEW E 98, 052415 (2018)

FIG. 1. Volvox and its inversion. (a) Volvox spheroid with somaticcells and one embryo labeled. Light microscopy image by StephanieHöhn reproduced from Ref. [15]. Scale bar: 50 μm. (b) Midsagittalcross section of the cell sheet, illustrating the sequence of cell shapechanges driving inversion in Volvox carteri, following Refs. [11,13]:Initially, cells are spindle-shaped [S]. Inversion starts as a bendregion of flask-shaped cells [F] with thin stalks forms, connectedby cytoplasmic bridges (CBs) at their stalks. As cells exit the bendregion, they become columnar [C]. Red line marks position of CBs;arrow marks direction of propagation of the bend region of flask-shaped cells. (c) Inversion types, reproduced from Ref. [16]. Toprow: type-A inversion; bottom row: type-B inversion. Labels “ant”and “post” indicate anterior and posterior hemispheres. P, phialopore;L, lip; I, invagination. Arrow indicates the course of time.

derive an averaged elastic model for the lips and we relate themechanical observations to the experimental results for Volvoxcarteri referenced above.

II. ELASTIC MODEL

The elastic model builds on the model that we have de-rived previously to describe type-B inversion in detail [17],although the present calculation is more intricate becauseaxisymmetry is broken owing to the presence of the lips.We consider a spherical shell of radius R and uniform thick-ness h � R [Fig. 2(a)], characterized by its arclength s anddistance from the axis of revolution ρ(s). Cuts in planescontaining the axis of symmetry divide part of the shell intoN lips of angular extent 2ϕ = 2π/N , as shown in Fig. 2(b).

A. The differential geometry of lips

We start by considering a single lip, −ϕ � φ � ϕ, whereφ is the azimuthal angle of the undeformed sphere. Comparedto an azimuthally complete shell, the cuts allow an additionaldeformation mode of the shell, one of azimuthal compressionor expansion. Here, we restrict to the simple deformation ofuniform stretching or compression so that the azimuthal anglein the deformed configuration of the shell is

φ = �(s)φ. (1)

FIG. 2. Elastic model. (a) Undeformed geometry: A sphericalshell of radius R and thickness h � R is characterized by itsarclength s and distance from the axis of revolution ρ(s ). (b) Anteriorview of lips: Cuts in planes containing the axis of symmetry defineN lips extending over −ϕ � φ � ϕ. (c) Deformed configuration:The midplane φ = 0 of the lip is characterized by its arclengthS(s ) and distance r (s ) from the axis of revolution. A local basis(ur , uφ, uz ) describes the deformed surface. (d) Deformation oftwo lips under the geometric simplification (1). The circumferentialcurvature changes sign at the point where the lip is perpendicular tothe axis of revolution.

In the deformed configuration, the distance from the axisof revolution is r (s), and the coordinate along the axis isz(s) [Fig. 2(c)]. These are assumed to be independent of φ.This is a geometric simplification that nonetheless ensurescoupling of the meridional and circumferential deformations;we discuss the basis for this approximation in more detailin Appendix A. With this simplification, points on the lipinitially at the same distance from and along the axis ofrevolution remain at the same distance from and along the axisof revolution as the lip deforms [Fig. 2(d)].

The deformed arclength, however, is a function of both s

and φ, and we define S(s) to be the arclength of the midlineφ = 0 of the lip. The meridional and circumferential stretchesof the midline of the lip are

fs (s) = dS

ds, fφ (s) = r (s)

ρ(s). (2)

The position vector of a point on the midsurface of thedeformed shell is thus

r (s, φ) = r (s)ur (�(s)φ) + z(s)uz, (3)

in a right-handed set of axes (ur , uφ, uz ) and so the tangentvectors to the deformed midsurface are

es = r ′ur + r�′φ uφ + z′uz, eφ = r� uφ, (4)

where dashes denote differentiation with respect to s. Bydefinition, r ′2 + z′2 = f 2

s , and so we may write

r ′ = fs cos β, z′ = fs sin β. (5)

The metric of the midsurface is thus{f 2

s + r2�′2φ2}ds2 + r2�2 dφ2 + 2r2��′φ ds dφ, (6)

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and its second fundamental form is

{fsβ′ + r�′2φ2 sin β} ds2 + r�2 sin β dφ2

+ 2r��′φ sin β ds dφ. (7)

Further, the unit normal to the deformed midsurface is

n = cos β uz − sin β ur . (8)

The Weingarten relations [2] yield

∂n∂s

= −κses + �′φ�

(κs − κφ )eφ,∂n∂φ

= −κφeφ, (9)

wherein

κs = β ′

fs

, κφ = sin β

r, (10)

are the principal curvatures of the midline of the lip. Hence theprincipal curvatures of the lip are those of its midline, thoughthe directions of principal curvature no longer coincide withes and eφ away from the midplane φ = 0 of the lips.

B. Calculation of the elastic strains

To calculate the deformation gradient, we make the Kirch-hoff “hypothesis” [20], that the normals to the undeformedmidsurface remain normal to the midsurface in the deformed

configuration of the shell. Taking a coordinate ζ across thethickness of the shell, the position vector of a general point inthe shell is

r (s, φ, ζ ) = rur + zuz + ζ n. (11)

The tangent vectors to the shell are thus

es = fs (1 − κsζ )(cos β ur + sin β uz )

+ ρ�′φfφ (1 − κφζ )uφ, (12a)

eφ = ρ�fφ (1 − κφζ )uφ . (12b)

The geometric deformation gradient tensor is therefore

Fg =(

fs (1 − κsζ ) 0

ρ�′φfφ (1 − κφζ ) �fφ (1 − κφζ )

). (13)

Cell shape changes impart intrinsic stretches f 0s , f 0

φ andcurvatures κ0

s , κ0φ to the cell sheet that are different from

its undeformed stretches and curvatures, but the cell shapechanges do not lead to any intrinsic azimuthal compression.Hence we define the intrinsic deformation gradient tensor

F0 =(

f 0s

(1 − κ0

s ζ)

00 f 0

φ

(1 − κ0

φζ)), (14)

Invoking the standard multiplicative decomposition of mor-phoelasticity [21], the elastic deformation gradient tensor F isdefined by Fg = FF0. Hence the Cauchy–Green tensor is

C = F�F =

⎛⎜⎜⎜⎜⎝

[fs (1 − κsζ )

f 0s

(1 − κ0

s ζ)]2

+ ρ2�′2φ2

[fφ (1 − κφζ )

f 0s

(1 − κ0

s ζ)]2 ρ�′�φf 2

φ (1 − κφζ )2

f 0s f 0

φ

(1 − κ0

s ζ)(

1 − κ0φζ

)ρ�′�φf 2

φ (1 − κφζ )2

f 0s f 0

φ

(1 − κ0

s ζ)(

1 − κ0φζ

) [�fφ (1 − κφζ )

f 0φ

(1 − κ0

φζ) ]2

⎞⎟⎟⎟⎟⎠, (15)

from which we derive the elastic strains ε = 12 (C − I). While we do not make any assumptions about the geometric or intrinsic

strains associated with Fg or F0, respectively, we assume that these elastic strains remain small. We therefore approximate thestrains along the midline by

fs (1 − κsζ ) ≈ f 0s

(1 − κ0

s ζ), fφ (1 − κφζ ) ≈ f 0

φ

(1 − κ0

φζ)/

�, (16)

except in differences of these expressions, so that

εss ≈ fs (1 − κsζ )

f 0s

(1 − κ0

s ζ) − 1 + 1

2ρ2� ′2φ2

[f 0

φ

(1 − κ0

φζ)

f 0s

(1 − κ0

s ζ)]2

, εφφ ≈ �fφ (1 − κφζ )

f 0φ

(1 − κ0

φζ) − 1, εsφ ≈ εφs ≈ 1

2ρ� ′φ

f 0φ

(1 − κ0

φζ)

f 0s

(1 − κ0

s ζ) ,

(17)

where we have introduced � = log �.

C. Calculation of the elastic energy

To derive the elastic energy, we need to specify the constitutive relations. As in our previous work [15–17], we assume that theshell is made of a Hookean material [20,22], characterized by its constant elastic modulus E and its Poisson ratio ν. The elasticmodulus appears only as an overall constant that ensures that E has units of energy. We shall assume moreover that ν = 1/2 forincompressible biological material. The elastic energy (per unit extent in the meridional direction) is thus

E2πρ

= E

2(1 − ν2)

∫ h/2

−h/2

{1

2ϕ

∫ ϕ

−ϕ

[(εss + εφφ )2 − 2(1 − ν)(εssεφφ − εsφεφs )

]dφ

}dζ . (18)

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PIERRE A. HAAS AND RAYMOND E. GOLDSTEIN PHYSICAL REVIEW E 98, 052415 (2018)

Performing the integrals across ζ and φ and expanding up to and including third order in h, we obtain

E2πρ

= Eh

2(1 − ν2)

{E2

s + E2φ + 2νEsEφ + 2aλ2� ′2(Es + νEφ ) + a(1 − ν)λ2� ′2 + 9

5a2λ4� ′4

}

+ Eh3

24(1 − ν2)

{K2

s + K2φ + 2νKsKφ + 3κ0

s

2E2

s + 3κ0φ

2E2

φ + 2ν(κ0

s

2 + κ0s κ0

φ + κ0φ

2)EsEφ − 4κ0

s EsKs

− 4κ0φEφKφ − 2ν

(κ0

s + κ0φ

)(EsKφ + EφKs

) + 2aλ2� ′2[(6κ0s

2 − 6κ0s κ0

φ + κ0φ

2)Es

+ νκ0s

(3κ0

s − 2κ0φ

)Eφ − (

3κ0s − 2κ0

φ

)Ks − ν

(2κ0

s − κ0φ

)Kφ

]+ a(1 − ν)λ2� ′2(3κ0

s

2 − 4κ0s κ0

φ + κ0φ

2) + 9

5a2λ4� ′4(10κ0

s

2 − 16κ0s κ0

φ + 6κ0φ

2)}, (19)

wherein a = π2/6N2 and λ = ρf 0φ /f 0

s , and the shell strains and curvature strains are defined by

Es = fs − f 0s

f 0s

, Ks = fsκs − f 0s κ0

s

f 0s

, Eφ = �fφ − f 0φ

f 0φ

, Kφ = �fφκφ − f 0φ κ0

φ

f 0φ

. (20)

We derive the governing equations associated with this energyin Appendix B. We solve these equations numerically usingthe boundary-value problem solver bvp4c of MATLAB (TheMathWorks, Inc.) and the continuation software AUTO [23].

III. RESULTS

A. Estimates of model parameters

We now specialize our model to describe type-A inversionin Volvox carteri by, in particular, encoding the observedcell shape changes [Fig. 1(b)] into the functional forms ofthe intrinsic stretches and curvatures. We shall introduce anumber of parameters for this purpose; we base our estimatesof these parameters on the measurements in Table I. Someof the values in Table I are taken from the literature, othersare extracted from figures in the literature. In particular, theseextracted values should not be taken as estimates of averagevalues but rather as indications of which values can be realizedexperimentally.

To describe the geometry of the undeformed shell[Fig. 3(a)], we must specify the number N of lips, the relative

TABLE I. Measurements of geometric parameters for type-Ainversion in Volvox carteri from previous measurements or extractedfrom previously published experimental figures. BR: bend region.

Quantity Measurement Reference

Number of lips N 4 [10]; [13], Fig. 2(c)Radius Ra ∼35 μm [10], Fig. 8(b); [13], Fig. 2(f)Thickness hb ∼7 μm [13], Fig. 2(f), Table 1Cell widthsc

Spindle cells 2.14 μm [13], Table 1Flask cells 2.05 μm Calc. from Ref. [13], Table 1Columnar cells 2.36 μm [13], Table 1

Phialopore P ∼0.3 [13], Fig. 2(f); [18], Fig. 1(b)Fraction of cells in BR ∼0.25 [11], Figs. 2(c)–2(e)

aRadius of cell sheet at phialopore opening.bThickness of cell sheet at phialopore opening.cCell widths are estimated along the midline of the cell sheet.

thickness h/R of the cell sheet, the opening angle P of thephialopore, and the extent of the lips. In accordance with themeasurements in Table I, we choose

N = 4, h/R = 0.21, P = 0.3, = 0.7. (21)

The estimate of is not based on a measured value, since itis rather hard to visualize the precise extent of the lips, butit is in qualitative agreement with experimental visualizationsof the lips [8,10,11]. Lips are clearly visible before inversionstarts [8], but additional breaking of cytoplasmic bridgescould increase during inversion. While breaking of cytoplas-mic bridges was suggested as a possible mechanism to explainthe cell rearrangements observed near the phialopore in type-B inversion [17], what experimental data there are [8,10,11]suggest that this effect is at most small in type-A inversion,

FIG. 3. Definition of model parameters. (a) Undeformed con-figuration after formation of spindle-shaped cells: The phialoporehas opened by an angle P on a spherical shell of radius R andthickness h. The lips span an angle . (b) Functional forms ofthe intrinsic stretches f 0

s , f 0φ , as functions of the polar angle θ .

Parameters fflask and fcol define the intrinsic stretches of the flask-and column-shaped cells relative to the spindle-shaped cells. Thewidth of the bend region is w, and the posterior limit of the bendregion is at θ = �. (c) Functional forms of the intrinsic curvaturesκ0

s , κ0φ , as functions of the polar angle θ and in units where R = 1.

The intrinsic meridional curvature in the bend region is −κflask. Fornumerical convenience, discontinuities in the intrinsic stretches andcurvatures are regularized over a small angular extent �θ = 0.1.

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FIG. 4. Bifurcation behavior of the lips. Diagrams in (�, E ) space of some solution branches for (a) k = 4.0 > k∗ and (b) k = 2.5 < k∗.Roman numerals label branches. The thickest, red line (branch I) corresponds to the branch of lowest energy connected to the initial state of theshell. Dashed lines in (b) indicate branches that are not connected to the lowest-energy branch. Insets show some solution shapes. On dottedparts of branches, solution shapes self-intersect.

justifying the absence of a “fracture criterion” for cytoplasmicbridges in the model.

The remaining parameters describe the functional forms ofthe intrinsic stretches and curvatures of the shell [Figs. 3(b)and 3(c)]: From measurements of the cell widths (Table I), weestimate the stretches fflask and fcol corresponding to flask andcolumnar cells (relative to spindle-shaped cells). The width w

of the bend region can be estimated from the fraction of flask-shaped cells in a midsagittal cross section of the cell sheet(Table I). Our estimates for these parameters are therefore

fflask = 0.95, fcol = 1.1, w = 0.65. (22)

Note that we may only read the actual stretches, as opposedto the intrinsic stretches, off the deformed shapes, but sincestretching is energetically more costly than bending, we ex-pect the approximations involved in obtaining these parameterestimates from cell size measurements to be good.

We do not estimate the final parameter, κflask, the intrinsicmeridional curvature of the flask cells, which is the mainparameter that we vary in the analysis that follows.

B. To flip or not to flip

We fix the value of k = κflask and, starting from a nearlyundeformed shell, propagate the bend region from the tip ofthe lips to their base and then towards the posterior pole bydecreasing the value of parameter � [Figs. 3(b) and 3(c)]that describes the position of the bend region. This quasistaticapproach to the evolution of the position of the bend region isjustified by the slow speed of the cell shape changes: Inversionin Volvox carteri takes about 45 min to complete [11].

Solution branches in (�, E ) space are shown in Fig. 4;there is a critical value k∗ separating two kinds of behavior. Ifk > k∗, then the shell inverts on the branch of lowest energy[Branch I in Fig. 4(a)]. Several branches bifurcate off the latter[Branches II–IV in Fig. 4(a)], but these have higher energy. Ifk < k∗, then the shell does not invert on the branch of lowestenergy or the branch connected to it [Branches I and II in

Fig. 4(b)]. There do exist branches on which the shell inverts[Branches III–V in Fig. 4(b)], analogous to those in Fig. 4(a),but these are not connected to the initial state of the shell.The topology of these additional branches undergoes anotherbifurcation, not discussed here, as k is reduced further. Whilewe might expect the global energy minimizers in Fig. 4 tobe stable equilibria, we do not analyze the stability of thedifferent equilibria on the various branches further in thispaper.

Some solution shapes on the branches in Fig. 4 self-intersect; we expect the corresponding parts of the branches tobe replaced with configurations of the shell where the rim ofthe lips is in contact with the uninverted part of the cell sheet.In these configurations, axisymmetry is necessarily brokenin the uninverted part of the shell; we do not pursue thisfurther, although we note that we have previously analyzed ananalogous contact problem in the absence of lips [17]. Theseconfigurations will not in fact be important for the discussionthat follows. Finally, we note that no such self-intersectingconfigurations arise on Branch III in Fig. 4(a), the solutionson which do lead to a completely inverted shell.

The flip-over of the lips on Branch I is illustrated inFig. 5: As k is reduced towards k∗, the lips open wider andwider before they flip over; after flip-over, the opening of thephialopore decreases quickly. As k is reduced below k∗, themaximal opening of the lips decreases; they do not flip overand the phialopore remains wide open.

Nishii et al. [19] showed that the InvA mutant of Volvoxcarteri fails to invert. In this mutant, there is no relative motionbetween cells and cytoplasmic bridges, and so the flask-shaped cells are not connected at their thin tips only [19]. Thusthe splay imparted, in the wild type, by the combination of cellshape change and motion of cytoplasmic bridges is reduced.This corresponds, in our model, to the intrinsic curvature κflask

being reduced in the InvA mutant. The mechanical bifurcationdiscussed above can thus rationalize the failure of the mutantto invert. The sequence of shapes on Branch I of Fig. 4(b) isindeed in excellent qualitative agreement with that observed

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PIERRE A. HAAS AND RAYMOND E. GOLDSTEIN PHYSICAL REVIEW E 98, 052415 (2018)

FIG. 5. Flip-over of lips. Radius of the phialopore rphial (nor-malized with initial phialopore radius sin P ) against � on BranchI for different values of k = κflask. On branches with k > k∗ (thicklines), self-intersecting solution shapes (dotted lines) arise. No suchself-intersections arise on branches with k < k∗ (thin lines).

during “inversion” of the InvA mutant, shown in Fig. 1(f)of Ref. [19]: The lips begin to curl over, but as “inversion”progresses, the lips do not flip over and the phialopore remainswide open at the end of inversion.

C. The importance of being contracted

We are left to discuss the observations of Nishii andOgihara [18], who showed that inversion of the Volvox carteriembryo is arrested if actomyosin-mediated contraction is in-hibited by various chemical treatments. They argued that it isthe resulting lack of contraction of the spindle-shaped cells inthe posterior, i.e., the relative expansion of the inverted partof the cell sheet, where the cells are columnar [Fig. 1(b)],that arrests inversion, the posterior hemisphere being swollencompared to the inverted part of the cell sheet. We thereforemodel actomyosin inhibition by setting fflask = fcol = 1. Thismodification does not, however, increase the critical curvaturek∗ very much (Fig. 6). Accordingly, if the arrest of inversion ofthe treated embryos were solely caused by this lack of relativeexpansion, then it would follow that inversion operates quiteclose to its mechanical limit.

There is, however, a curious observation that appears,almost as a footnote, in the caption of Fig. 6 of Ref. [18]:In embryos in which contraction had been inhibited, thenumber of cells constituting the bend region was smaller thanin untreated embryos. While it is unclear why the chemicaltreatments applied in Ref. [18] should have this effect, it canbe introduced into the model by reducing the value of w. Toestimate the magnitude of this effect very roughly, we turn topreviously published experimental figures: In untreated em-bryos, about seven cells make up the bend region (Fig. 2(c)–2(e) in Ref. [11]); in treated embryos, this is reduced toabout four (Fig. 6(d) and 6(e) in Ref. [18]). We thereforeestimate w ≈ 0.37 in the treated embryos. With this valueof the width of the bend region, the critical curvature k∗ isincreased considerably (Fig. 6), suggesting that inversion does

FIG. 6. Effect of inhibiting contraction. Critical curvature k∗against parameter sets: Open circles indicate values identical to theestimated “wild-type” values (WT); filled circles indicate modifiedparameter values. Modified parameter values corresponding to thechemical treatments of Ref. [18] are fflask = fcol = 1, w = 0.37.

not need to be close to its mechanical limit to explain theobserved arrest of inversion.

Moreover, cells in the bend region of the treated embryosare less markedly flask-shaped than those in the untreatedones (Fig. 6 in Ref. [18]), which might indicate that theintrinsic curvature κflask is reduced in the untreated embryos.This reduction of the intrinsic curvature may provide anotherexplanation for the failure of actomyosin-inhibited embryosto invert. The experimental images in Ref. [18] suggest that,as in “inversion” of the InvA mutant [19], the lips start topeel back but then fail to flip over completely, as in theshapes obtained in the model for low values of the intrinsiccurvature [Fig. 4(b)].

Closer examination of the shapes of the treated embryosin Ref. [18] suggests that the treated embryos are crammedinto the embryonic vesicle that surrounds the embryos duringinversion because of lack of contraction of the spindle-shapedcells. In fact, Ueki and Nishii [24] studied the InvB mutant ofVolvox carteri in which the embryonic vesicle fails to growproperly during development. They showed that inversionis prevented in the InvB mutant by the confining forces ofthe embryonic vesicle: Inversion completes if and only ifthe InvB mutant is microsurgically removed from the em-bryonic vesicle [24]. Nishii and Ogihara [18] reported thatfragments of treated embryos removed from the embryonicvesicle can invert but left open the question whether completetreated embryos can invert when removed from the embryonicvesicle. While the above discussion suggests that lack ofrelative expansion is not the mechanical reason for inversionfailure in the treated embryo, this experiment could help todecide which of the three other candidate mechanisms is thedominant cause of inversion arrest: Is it the reduction of thewidth of the bend region, the reduced intrinsic curvature inthe bend region, or the confinement of a swollen embryo tothe stiff embryonic vesicle that prevents the lips from flippingover completely? The final effect could also play a role inthe InvA mutant since, as discussed previously, the maximal

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FIG. 7. Effective geometry for lips. (a) Geometric considera-tions: A lip of length = R(L − P ) on a shell of radius R hasfolded into an arc of negative curvature and of radius R̂ interceptingan angle 2χ , rotated by an angle ϑ with respect to the originalconfiguration. The chord intercepted by the lip makes an angle ψ

with the axis of the shell. (b) Details for the calculation of thedisplacement of the midpoint of the lip.

opening of the phialopore increases as the intrinsic curvatureis reduced above the critical curvature (Fig. 5).

D. Effective energy

To gain some insight into the physical mechanism underly-ing the flip-over of the lips, it is useful to consider a reduced(two-parameter) model that balances three physical effects:

(i) the stretching energy associated with the hoop stretchesinduced by the bending of the lips;

(ii) the bending energy associated with deviations of thecurvature of the bent lips from its intrinsic value;

(iii) the elastic energy of the formation of a second bendregion that links the bent lips up to the remainder of the shell.

We begin by describing the reduced geometry: We consideran elastic spherical shell of undeformed radius R, with aphialopore of angular extent P at its anterior pole [Fig. 7(a)].Cuts define N lips of length = R(L − P ) adjacent to thisopening. As the shell deforms, these lips bend into circulararcs of radius R̂ (and negative curvature), intercepting anangle 2χ . Since stretching is energetically more costly thanbending, there cannot be any stretching at leading order, andthus 2χR̂ = . In what follows, we nondimensionalize lengthswith R and energy densities with Eh.

Further, the base of the lips may rotate by an angle ϑ withrespect to the undeformed configuration; as a result, the chordintercepted by the lip makes an angle ψ with the axis of theshell, where ψ = L − (90◦ − χ + ϑ ). The distance from thetip of the deformed lip to the axis of the shell is thus

rP = sin L − 2R̂ sin χ cos (L + χ − ϑ ). (23)

Since we have already imposed that the meridional strainsvanish globally, the no-stress condition at the free edge of thelips forces the hoop strain there, EP = �P rP / sin P − 1, tovanish at leading order. Rearranging, the azimuthal compres-sion at the phialopore is thus

�P = sin P

sin L − 2R̂ sin χ cos (L + χ − ϑ ), (24)

and we let �P = log �P . At the base of the lip, �B = 1 andthus �B = 0 to match up to the part of the shell withoutlips. Hence we approximate � ′ = (�P − �B )/ = log �P /

at lowest order; this minimizes the integral of � ′2 along thelips. This assumption of constant � ′ also implies that, at themidpoint of the lip, log �M = �M = 1

2�P = 12 log �P , and

thus �M =√

�P .To describe the additional hoop strains resulting from the

bending of the lips, we compute the distance of the midpointof the deformed lip from the axis of revolution [Fig. 7(b)],

rM = sin L + R̂[sin (L − ϑ ) − sin (L + χ − ϑ )]. (25)

The hoop strain is therefore EM = �MrM/ρM − 1 at themidpoint of the lips, with ρM = sin (P + /2).

The effective elastic energy is F = F1 + F2 + F3, the sumof the three respective contributions of the physical effectsdescribed above. The stretching and bending energies F1,F2

of the deformed lips are the integrals over the lips of, respec-tively, the squared hoop strain E2

M and the squared curvaturestrain K2 = ε2(1/R̂ − 1/R0)2, where ε is the nondimensionalbending modulus and R0 is the intrinsic radius curvature of thelips. An estimate of the energy F3 associated with rotating thebase of the lips by ϑ can be obtained from the energetics of aPogorelov dimple [25]. Thus

F1 = ρM E2M, F2 = ρM ε2

(1

R̂− 1

R0

)2

, F3 = ε3/2ϑ2.

(26)

In these expressions for F1,F2, the factor ρM correspondsto integration over the length of the lips. Two parametersdetermine this reduced energy: the radius R̂ of the deformedlip and the angle ϑ .

We determine minima of F numerically using MATHEMAT-ICA (Wolfram, Inc.). Denoting by k0 = 1/R0 and k = 1/R̂

the intrinsic and actual curvatures of the bend region, weplot the position of the energy minima in the (k0, ϑ ) and(k0, k) diagrams (Fig. 8), for different values of the lip extent . At large-enough values of k0, k > k0 and ϑ < 0, so thelips bend and rotate outwards and thus flip over. At smallvalues of k0, ϑ > 0 and k < k0: The lips resist bending andflipping over by rotating inwards to alleviate hoop strains.We also note that a critical value ∗ separates two kinds ofbehavior: If < ∗, then the transition between the two statesis continuous but becomes discontinuous if > ∗, with thetwo states coexisting in an intermediate range of k0. Given theexistence of other solution branches discussed previously, thisbehavior is not surprising; the discontinuous transition signalsa breakdown of the geometric approximation of uniformlycurved lips as the lip extent grows.

Nonetheless, this discussion shows how the behavior ob-served in the continuum model can be attributed to threesimple physical effects. Conversely, if any of these threeeffects is not considered, the reduced model fails to capturethe observed behavior: Clearly, if F2 is neglected, there is nodependence on R0, and if F1 is neglected, there is no couplingbetween R̂ and ϑ , and there is a minimum R̂ = R0, ϑ = 0, forall R0. Finally, if the contribution of F3 is not considered, wefind, numerically, two minima with R̂ = R0 and ϑ = 0 for allR0, so that the transition between the two kinds of behavior

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PIERRE A. HAAS AND RAYMOND E. GOLDSTEIN PHYSICAL REVIEW E 98, 052415 (2018)

FIG. 8. Effective energy of lips. Coordinates of the minimumof the effective energy F in the (k0, ϑ ) diagram for parametervalues 1 = 0.37 < ∗, 2 = 0.55 > ∗, illustrating flip-over of lipsat large enough values of k0. Folds arise for > ∗; dotted lines markpositions of unstable saddle points. Inset: corresponding plot in the(k0, k) diagram.

discussed above is not reproduced. Hence all three of thesephysical effects are essential to explain the observed behavior.

IV. CONCLUSION

In this paper, we have derived a simple, averaged elastictheory to analyze the flip-over of lips observed during type-A inversion in Volvox carteri and have shown by meansof a reduced model how the observed behavior can be at-tributed to three geometrical effects. The model can explainthe observations of Nishii et al. on the InvA mutant [19] butsuggests that the failure of embryos treated with actomyosininhibitors to invert [18] does not result from lack of rela-tive expansion of the cell sheet. Several potential candidatemechanisms for the inversion arrest in the treated embryosremain, however. Further experiments on embryos removedfrom the embryonic vesicle could help to distinguish betweenthese three mechanisms discussed above and in particularclarify the role of the confinement to the embryonic vesiclefor both the chemically treated embryos and those of the InvAmutant.

Further experiments could also allow taking the present,qualitative analysis to a more quantitative level. The large sizeof the gonidia of Volvox carteri at the inversion stage mayhamper such a comparison between theory and experiment.This difficulty could be addressed by using the gonidialessmutants of Volvox carteri that invert normally [26] or othertype-A inverters such as Volvox gigas that have smaller goni-dia at the inversion stage [27].

It also remains unclear whether N = 4 lips are mechani-cally optimal for inversion in some sense. In the much moredynamic process of impact petalling of metallic sheets [28],during which lips similar to those seen in inversion arise, thenumber of lips is set by minimizing dissipation of elastic andplastic energy. It is tempting to argue that a similar mechanicaltrade-off arises for inversion: The more lips there are, the

easier they are to invert, but the coupling of the lips to theremainder of the shell and hence the extent to which the lipsaid inversion of the connected part of the shell decrease withincreasing lip number. While this mechanical argument doessuggest an intermediate optimal lip number, it ignores therather more combinatorial constraint that cells must divide ina way that robustly defines the lips. A very recent study [29]of a related cell packing problem in Drosophila egg chambershighlighted the role of entropic effects in selecting the spatialstructure of connected and unconnected adjacent cells. InVolvox carteri, the four lips are defined around the 16-cellstage [8]. It is tempting to speculate that this number of lipsis stabilized by similar entropic effects, especially since alarger lip number could only be defined at later stages of celldivision, with a combinatorial explosion of possible packings.The cell division pattern of Volvox carteri has been mapped upto and including the 64-cell stage [8]. Extending this work tolater cell division stages could shed more light on these issues.

All the algae of the family Volvocaceae display somekind of inversion [6,10], although, interestingly, the genusAstrephomene of the closely related family Goniaceae formsspherical colonies of up to 128 cells without the need forinversion [30]. Our analysis should therefore finally be con-sidered in the context of the evolution of Volvocaceae. Thepresent analysis of type-A inversion in Volvox carteri indicatesthat relative expansion of different parts of the cell sheetis not mechanically required for this inversion (although itmay play a role in enabling inversion within the confinementof the embryonic vesicle). By contrast, we have previouslyshown that the peeling of the anterior hemisphere [Fig. 1(c)]during type-B inversion in Volvox globator is mainly drivenby contraction of parts of the cell sheet [17]. This contractionis regulated separately from the earlier invagination of thecell sheet and therefore indicative of a transition towardshigher developmental complexity within Volvocaceae [17].The present result that type-A inversion does not rely on thisadditional deformation mode lends further support to thisinference. This additional complexity in type-B inversionthus appears as the geometric price of the absence of lips.The questions how these different features—formation of lipsand separately regulated inversion subprocesses—evolved,and in particular, whether they were lost from an ancestralalga, remain widely open. The recent observation that inver-sion in the genus Pleodorina features nonuniform cell shapechanges [31], shared with type-B inversion [14] but absentfrom type-A inversion, might begin to shed some light onthese issues.

ACKNOWLEDGMENTS

We thank Stephanie Höhn for many useful discussionsabout the biology of inversion and comments on a draftof this paper and are grateful for support from the Engi-neering and Physical Sciences Research Council (EstablishedCareer Fellowship EP/M017982/1, R.E.G.; Doctoral PrizeFellowship, P.A.H.), the Schlumberger Chair Fund, the Well-come Trust (Investigator Award 207510/Z/17/Z, R.E.G.), andMagdalene College, Cambridge (Nevile Research Fellowship,P.A.H.).

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APPENDIX A: GEOMETRIC SIMPLIFICATIONS

In this Appendix, we discuss the geometric approximationsin more detail. For a general geometric description of the lips,we must replace Eq. (3) with

r (s, φ) = r (s, φ) ur (φ(s, φ)) + z(s, φ) uz. (A1)

By symmetry, φ is an odd function of φ, while r and z areeven functions of φ. Expanding,

r (s, φ) = r (s) + R(s)φ2 + O(φ4), (A2a)

φ(s, φ) = �(s)φ + O(φ3), (A2b)

z(s, φ) = z(s) + Z(s)φ2 + O(φ4). (A2c)

In particular, merely on grounds of symmetry, we thus expectvariations of φ across the lips to be of order O(ϕ) = O(N−1).Hence, at least in the limit of a large number of lips, thesevariations swamp those of r and z, which result from R andZ and are thus expected to be of order O(ϕ2) = O(N−2). Theargument thus suggests that the largest deformations are thoseof simple azimuthal compression, φ(s, φ) = �(s)φ, to whichwe have restricted at the start of our analysis.

It is, however, important to note that this does not implythat the elastic theory we have derived is the asymptotic limitof the general theory in the limit of a large number of lips. Itis not hard to see that in fact, a theory that is asymptoticallyexact to order O(ϕ2) depends on the corrections to r andz up to order O(ϕ4). Despite the geometric restriction thatthe simple azimuthal compression that we have imposedin Eq. (1) therefore implies, we note that the theory thatresults from it and that we have analyzed in this paper issimple enough to allow some detailed analysis, yet features acrucial coupling between the meridional and circumferential

deformations resulting from �. The importance of this cou-pling is revealed by the observation that, in its absence, the lipscan zero their contribution to the energy density by adoptingtheir intrinsic meridional stretches and curvatures (as one-dimensional elastic filaments would do) and then compressingazimuthally at no energetic cost to make the circumferentialstrain and curvature strain vanish. This decouples the lips fromthe remainder the shell, so their flipping over does not help theremainder of the shell to invert.

APPENDIX B: GOVERNING EQUATIONS

In this Appendix, we derive the governing equations asso-ciated with the elastic energy (19). We note the variations

δEs = 1

f 0s

(sec β δr ′ + fs tan β δβ ), (B1a)

δEφ = 1

f 0φ

(fφ� δ� + �

ρδr

), (B1b)

and

δKs = δβ ′

f 0s

, (B1c)

δKφ = 1

f 0φ

(fφκφ� δ� + �

cos β

ρδβ

), (B1d)

which are obtained from the definitions of the strains. Thevariation of the elastic energy takes the form

δE2πρ

= nsδEs + nφδEφ + msδKφ +mφδKφ + �

ρδ� ′, (B2)

wherein the shell stresses are

ns = Eh

1 − ν2(Es + νEφ + aλ2� ′2) + Eh3

12(1 − ν2)

[3κ0

s

2Es + ν

(κ0

s

2 + κ0s κ0

φ + κ0φ

2)Eφ − 2κ0

s Ks

− ν(κ0

s + κ0φ

)Kφ + aλ2� ′2(6κ0

s

2 − 6κ0s κ0

φ + κ0φ

2)], (B3a)

nφ = Eh

1 − ν2(Eφ + νEs + νaλ2� ′2) + Eh3

12(1 − ν2)

[3κ0

φ

2Eφ + ν

(κ0

s

2 + κ0s κ0

φ + κ0φ

2)Es − 2κ0

φKφ

− ν(κ0

s + κ0φ

)Ks + aλ2� ′2κ0

s

(3κ0

s − 2κ0φ

)], (B3b)

the shell moments are

ms = Eh3

12(1 − ν2)

[Ks + νKφ − 2κ0

s Es − ν(κ0

s + κ0φ

)Eφ − aλ2� ′2(3κ0

s − 2κ0φ

)], (B3c)

mφ = Eh3

12(1 − ν2)

[Kφ + νKs − 2κ0

φEφ − ν(κ0

s + κ0φ

)Es − νaλ2� ′2(2κ0

s − κ0φ

)], (B3d)

and where

� = 2aρλ2� ′{

Eh

1 − ν2

(Es + νEφ + 1 − ν

2

)− Eh3

12(1 − ν2)

[(3κ0

s − 2κ0φ

)Ks + ν

(2κ0

s − κ0φ

)Kφ

− (6κ0

s

2 − 6κ0s κ0

φ + κ0φ

2)Es − νκ0

s

(3κ0

s − 2κ0φ

)Eφ − 1 − ν

2

(3κ0

s

2 − 4κ0s κ0

φ + κ0φ

2)]}

+ 18

5a2ρλ4� ′3

[Eh

1 − ν2+ Eh3

12(1 − ν2)

(10κ0

s

2 − 16κ0s κ0

φ + 6κ0φ

2)]. (B3e)

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PIERRE A. HAAS AND RAYMOND E. GOLDSTEIN PHYSICAL REVIEW E 98, 052415 (2018)

Finally, on letting

Ns = ns

fφf 0s

, Nφ = �nφ

fsf0φ

, Ms = ms

fφf 0s

, Mφ = �mφ

fsf0φ

, (B4)

the variation becomes

δ

(∫ E2π

ds

)= �rNs sec β δr + rMs δβ + � δ�� −

∫ [d

ds(rNs sec β ) − fsNφ

]δr ds

−∫ [

d

ds(rMs ) − rfsNs tan β − fsMφ cos β

]δβ ds −

∫ [d�

ds− rfs (Nφ + κφMφ )

]δ� ds, (B5)

wherein the final term that did not appear in our previous work [15,17] is the contribution from the lips. A factor of fs wasomitted from the corresponding variation in Ref. [15], but the results in the paper are correct. As in standard shell theories [22],we define the transverse shear tension T = −Ns tan β to remove a singularity in the resulting equations, so that

dNs

ds= fs

(Nφ − Ns

rcos β + κsT

),

dMs

ds= fs

(Mφ − Ms

rcos β − T

),

d�

ds= rfs (Nφ + κφMφ ). (B6a)

By differentiating the definition of T and using the first of Eqs. (B6a), one finds that

dT

ds= −fs

(κsNs + κφNφ + T

rcos β

). (B6b)

Together with the geometrical equations r ′ = fs cos β and β ′ = fsκs , Eqs. (B6) describe the deformed shell. The final shapeequation z′ = fs sin β is redundant. From the boundary terms in Eq. (B5), we deduce the seven pertaining boundary conditions:

r = 0, β = 0, T = 0, � = 1 =⇒ � = 0, (B7a)

at the posterior,

Ns = 0, Ms = 0, � = 0, (B7b)at the phialopore.

Remark on the numerical solution of Eqs. (B6)

At each stage of the numerical solution, fφ and κφ are computed directly from r and β, but it is less straightforward tocompute � ′, Es , Ks from �, Ns , Ms , required in order to obtain fs , κs , Nφ , Mφ and hence continue the integration. To this end,we eliminate Es , Ks among Eqs. (B3a), (B3c), and (B3e) by solving a linear system of equations to obtain a cubic equationfor � ′, which is solved exactly using the algorithm described in Ref. [32]. Once � ′ is known, Eqs. (B3a) and (B3c) become alinear system of equations for Es , Ks . From these, fs , κs , Nφ , and Mφ can be computed and the integration can be continued.

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